The B B Newman spelling theorem

Nyberg-Brodda, Carl-Fredrik

doi:10.1080/26375451.2021.1904735

Cited by 5 publications

(3 citation statements)

References 34 publications

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“…However, Pride reports on a counterexample to this statement, discovered by Newman. In fact, the example appears already in Newman's Ph.D. thesis (see [67] for the story of how this thesis was uncovered): where we use the convention that a b = bab −1 . In particular, as H is hyperbolic, being a finite index subgroup of a hyperbolic group (or applying Corollary 1.4), the Diophantine problem is decidable in the torsion-free one-relator group Gp a, b, c | a b a c = a .…”

Section: Torsion Subgroupsmentioning

confidence: 99%

On the Diophantine problem in some one-relator groups

Nyberg-Brodda¹

2022

Preprint

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We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister-Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation a m b n = 1 is virtually a direct product of hyperbolic groups for all m, n ≥ 0, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the twogenerator, one-relator groups Gm,n with defining relation a commutator [a m , b n ] = 1, where m, n ≥ 1. In doing so, we define and study a natural class of groups (rabsags), related to right-angled Artin groups (raags). We reduce the Diophantine problem in the groups Gm,n to the Diophantine problem in groups which are virtually certain rabsags. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group G 2,2 with the single defining relation [a 2 , b 2 ] = 1. We use the recent classification by Gray & Howie of raag subgroups of one-relator groups to classify the raag subgroups of some rabsags, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups NG(p, q), which are (p + 1)-generated one-relator groups generalising the solvable Baumslag-Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.

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Section: Torsion Subgroupsmentioning

confidence: 99%

On the Diophantine problem in some one-relator groups

Nyberg-Brodda¹

2022

Preprint

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“…For example, one cannot even decide if a finitely presented group is trivial or not! For more details, we refer the reader to the author's English translation [53] of the articles on this subject by Adian and Markov. Notably, Tseytin [72] also contributed to this subject, providing a slight extension (focussed on the number of generators).…”

Section: Encodingsmentioning

confidence: 99%

Correction to: The word problem for one-relation monoids: a survey

Nyberg-Brodda

2022

Semigroup Forum

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“…Semigroup presentations go back to the mid-twentieth century, and again have proven a crucial tool in the above fields. For some early studies, see for example [1,2,100,101], and for early papers with connections to logic see [84,102,116]; some historical information can be found in [97].…”

Section: Introductionmentioning

confidence: 99%

On a class of semigroup products

Carson¹,

Dolinka²,

East³

et al. 2022

Preprint

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This paper concerns a class of semigroups that arise as products U S, associated to what we call 'action pairs'. Here U and S are subsemigroups of a common monoid and, roughly speaking, S has an action on the monoid completion U 1 that is suitably compatible with the product in the over-monoid.The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure a comprehensive understanding of them. We then apply our abstract results to many special cases of interest.The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup U S is a quotient of a semidirect product U ⋊S, and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (a.k.a. E-unitary) inverse semigroups.The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for U S out of presentations for the constituents U and S in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.

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The B B Newman spelling theorem

Cited by 5 publications

References 34 publications

On the Diophantine problem in some one-relator groups

On the Diophantine problem in some one-relator groups

Correction to: The word problem for one-relation monoids: a survey

On a class of semigroup products

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